The ionization and generalized ionization conjectures

A nucleus can hold only a fixed number of extra electrons, however large its charge.

3 papers September 24, 2026 Explained by AI

A nucleus of charge Z holds Z electrons when its atom is neutral, and perhaps a few more. One paper claims to prove that the number of extra electrons stays below one fixed bound, for atoms and molecules. Two companion papers prove large-atom laws for the energy needed to remove many electrons. They also cover the size of the outer electron cloud. These two laws have Lean proofs, but the bound on extra electrons does not.

First step

1984

Elliott Lieb, bound linear in the nuclear charge

Significance

Major

It claims to settle a problem from Barry Simon's 2000 survey and two conjectured large-atom laws.

Trust

Formal proof listed

Field

Mathematical physics

3 papers

The explanation

Take a bare nucleus with charge 100. Put 100 electrons around it and you have a neutral atom. Can it hold more? An extra electron counts as bound if adding it lowers the lowest possible energy. The best earlier bound for atoms came in 2025. It allows fewer than 1.1185Z + 4Z^(1/3) electrons, where Z is the charge and Z is at least 4. For Z = 100 that is about 130.4, so up to 30 extra electrons. For Z = 10,000 it is about 11,271, so up to 1,271 extra. The allowance grows with the charge.

The first paper claims to prove that it does not need to grow. There is one fixed number C that works for every nucleus. A nucleus of charge Z strictly binds at most Z + C electronsSource 1. The same holds for molecules. With M fixed nuclei of total charge Z, the limit is Z + CMSource 1. Each nucleus needs charge at least 1, and the nuclei may sit anywhereSource 2. The theorem asserts that such a C exists without giving its value.

This is the ionization conjecture. Barry Simon posed its atomic form as Problem 9 in a 2000 survey. It asks whether the number of extra electrons stays bounded as the charge grows. In 1984, Elliott Lieb proved bounds that grow in proportion to the nuclear charge. Lieb, Sigal, Simon and Thirring showed that the ratio of bound electrons to charge tends to 1. Phan Thành Nam proved fewer than 1.22Z + 3Z^(1/3) in 2012. The 2025 bound above came from Hundertmark, Pattakos and Schulz. These results still left the fixed-allowance question open.

The setting is the full quantum model of an atom, without relativity. Nuclei are held fixed, electrons have two spin states, and every electron repels every other one. In 2003, Jan Philip Solovej proved uniform bounds in Hartree–Fock theory. That is an approximation that builds the electron state from single-electron pieces. The real model allows correlation, meaning where one electron sits changes where the others tend to beSource 3. That is the main obstacle. The outer electrons also sit far from the bulk of the atom. They are a vanishing fraction of it, so knowing the bulk density does not settle themSource 4. The papers handle correlation with a statistical device. They imagine blurry snapshots of all electron positions and study the state given those snapshotsSource 5.

For neutral atoms, the first paper makes two more claims. Removing one electron costs an energy between two fixed positive numbers, for every atomSource 6. That cost is called the first ionization energy. The sphere outside which half an electron remains, on average, also has a radius between two fixed boundsSource 7. In 1990, Seco, Sigal and Solovej had an upper bound on that energy, but it grew with Z.

In 2016, Solovej stated a generalized ionization conjecture that asks for exact laws, not just bounds. The second paper proves its energy part. Let I_m be the energy needed to strip m electrons from a neutral atom. Known large-atom formulas give each total energy only approximately. Subtracting two of them fails, because their error terms exceed the differenceSource 8. When m is large but small next to Z, I_m is close to a_TF × m^(7/3)Source 9. Here a_TF is a fixed positive constant from Thomas–Fermi theory, which treats the electrons as a continuous charged fluid. So doubling a large m multiplies the cost by about 2^(7/3), which is about 5.04. The paper gets this when m and Z/m both grow together. That is stronger than the order of limits in Solovej's statement, which it also obtainsSource 10.

The third paper proves the radius part. Draw a sphere around the nucleus so that, on average, m electrons lie outside it. Call its radius R_mSource 11. For heavy atoms and large m, R_m approaches b × m^(-1/3), with b = (81π²/2)^(1/3)Source 12. That is about 7.37 in atomic units of length. For m = 8 the law gives about 3.68. For m = 1,000 it gives about 0.74. Fewer electrons outside means a wider sphere. Solovej proved this same constant in 2003 for Hartree–Fock theorySource 13.

These laws describe limits, so they give no exact value for any single atom. In the radius law, the charge Z goes to infinity first, and m grows afterward. Neither law claims that the values settle down at a fixed mSource 14. All three papers concern the nonrelativistic model only.

How sure is this? A computer proof checker called Lean has checked formal versions of the two laws, for energy and for radius. Each has a reference statement for a matching tool called Comparator. The Lean scope note leaves out fixed-m convergence, and it takes the radius law's ground states as given. The note does not cover the first paper. So the bound on extra electrons and the first-ionization bounds rest on the written proof. No specialist review has been published.

What it doesn't show. This does not give a value for the constant C, or exact numbers for any particular atom.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major only if the proofs hold. Lean checks the two large-atom laws. The bound on extra electrons and the first-ionization bounds have no machine check, and no specialist review is published.

Words used

Nuclear charge: The positive charge of a nucleus, written Z; a neutral atom has Z electrons.
Strictly bound electron: An added electron that lowers the lowest possible energy of the system.
Ionization energy: The energy needed to remove one or more electrons from an atom.
Correlation: Where one electron sits changes where the others tend to be.
Thomas–Fermi theory: A classical model that treats the electrons as a continuous charged fluid.
Hartree–Fock theory: An approximation that builds the electron state from single-electron pieces.
Asymptotic law: A rule that becomes exact only in a limit, such as very large charge.

The original papers (3)

Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms (PDF, opens in a new tab)
September 24, 2026
Generalized ionization energies for full Coulomb atoms (PDF, opens in a new tab)
September 24, 2026 Formal proof listed
Generalized outer-electron radii of neutral Coulomb atoms (PDF, opens in a new tab)
September 24, 2026

Lean notes

All quoted lines (14)

There is a universal finite constant $C$ such that every nuclear system specified above and every integer $n\ge1$ satisfy $E_n<E_{n-1}\quad\Longrightarrow\quad n\le Z+CM.$
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
The positions may be arbitrarily close or far apart.
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
Those conclusions concern their stated variational models; the states in (…) may have arbitrary many-body correlations.
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
Its vanishing fraction makes bulk convergence insufficient to prove Theorem ….
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
To prove these comparisons, observe all positions with independent errors of width $\ell$ and forget the particle labels.
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
There are universal constants $0<c\le C<\infty$ such that the first ionization energy of the neutral atom satisfies $c\le I_1(Z):=E_Z(Z-1)-E_Z(Z)\le C \qquad\text{for every integer }Z\ge1.$
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
There are universal constants $0<c\le C<\infty$ such that, for every integer $Z\ge1$ and every normalized ground state of $H_{Z,Z}$ , $c\le R(\Psi_Z)\le C.$
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms Back to text
Subtracting two such total-energy asymptotics does not control a much smaller ionization energy: the remainder may exceed the difference being sought.
Generalized ionization energies for full Coulomb atoms Back to text
We prove its energy part for the full two-spin Coulomb Hamiltonian, in the stronger joint regime in which both the number removed and the ratio of nuclear charge to that number tend to infinity.
Generalized ionization energies for full Coulomb atoms Back to text
The joint assertion … strengthens that order of limits.
Generalized ionization energies for full Coulomb atoms Back to text
This is a radius in the neutral atom: $m$ specifies the expected electron mass outside the sphere.
Generalized outer-electron radii of neutral Coulomb atoms Back to text
Set $b_{\mathrm{TF}}=(81\pi^2/2)^{1/3}$ .
Generalized outer-electron radii of neutral Coulomb atoms Back to text
Solovej's unrestricted Hartree–Fock theorem established the universal outer-radius asymptotic with the same constant as in ….
Generalized outer-electron radii of neutral Coulomb atoms Back to text
No convergence of $I_m(Z)$ at a fixed $m$ is asserted or assumed.
Generalized ionization energies for full Coulomb atoms Back to text